Multi-Marginal Inverse Optimal Transport for Contrastive Learning Via Explicit Anchor-Positive-Negative Coupling

2026-09-27

AI Summary

This study addresses the dimensional collapse problem in existing inverse optimal transport (IOT) contrastive learning, which arises from neglecting negative samples. To this end, we propose a multi-margin inverse OT method that, for the first time, explicitly incorporates negative samples into the IOT framework. The approach optimizes representations through anchor-positive-negative triplet coupling and introduces an efficient push-pull algorithm to reduce computational complexity. Theoretically, we prove that the proposed method exhibits neural collapse properties and further present a low-complexity alternative. Extensive experiments on both synthetic and real-world datasets demonstrate that our method significantly outperforms existing OT-based contrastive learning approaches, effectively mitigating dimensional collapse while enhancing performance on downstream tasks.

Abstract

Inverse Optimal Transport (OT) based methods for representation learning learn representations such that the global OT coupling between a pair of data marginals in the representation space, concentrates on the positive pairs. This is in contrast to previous methods that primarily focused on pairwise matching. However, these methods $\textit{DO NOT}$ utilize negative pairs and hence are not truly contrastive in their approach. We show that this leads to issues of dimensional collapse and hence degraded downstream performance. To alleviate this, we develop a novel multi-marginal (MM) inverse OT (IOT) contrastive learning (CL) approach called Neg-MMIOT-CL, which learns representations such that the global multi-marginal OT (MMOT) coupling between a triple of data marginals, with respect to a carefully designed ground-cost between triplets of data points in the representation space, concentrates on the anchor-positive-negative $\textit{triplets}$. For a latent class model, we empirically show that Neg-MMIOT-CL alleviates dimensional collapse. Furthermore, for a specific choice of ground cost for all triplets in representation space, we prove that the optimal representation configuration for Neg-MMIOT-CL exhibits equiangular property for within-class and across-class representations, which translates to Neural-Collapse when the representation dimension is larger than the number of classes minus one -- a result that is $\textit{previously established only}$ for pairwise contrastive learning methods. Finally, we propose Neg-IOT-CL-PushPull, that is a computationally efficient alternative to Neg-MMIOT-CL, alleviating the high cost of computing MMOT plans needed during implementation. We apply these methods on both synthetic and real-world datasets and show significant improvements over existing OT-based contrastive learning methods.

Problem

• Inverse Optimal Transport
• Contrastive Learning
• Dimensional Collapse
• Representation Learning
• Multi-Marginal

Innovation

• Inverse Optimal Transport
• Multi-Marginal Contrastive Learning
• Dimensional Collapse
• Neural Collapse
• Anchor-Positive-Negative Coupling